Linear credit risk models ∗

Damien Ackerer † Damir Filipovic ‡

July 6, 2019

forthcoming in Finance and Stochastics

Abstract

We introduce a novel class of credit risk models in which the drift of the survival processof a firm is a linear function of the factors. The prices of defaultable bonds and credit defaultswaps (CDS) are linear-rational in the factors. The price of a CDS option can be uniformlyapproximated by polynomials in the factors. Multi-name models can produce simultaneous de-faults, generate positively as well as negatively correlated default intensities, and accommodatestochastic interest rates. A calibration study illustrates the versatility of these models by fit-ting CDS spread time series. A numerical analysis validates the efficiency of the option priceapproximation method.

Keywords: credit default swap, credit derivatives, credit risk, polynomial model, survival process

MSC (2010): 91B25, 91B70, 91G20, 91G40, 91G60

JEL Classification: C51, G12, G13

∗The authors would like to thank for useful comments Agostino Capponi, David Lando, Martin Larsson, JongsubLee, Andrea Pallavicini, Sander Willems, and two anonymous referees, as well as participants from the 2015 AMaMeFand Swissquote conference in Lausanne, the 2016 Bachelier world congress in New-York, the 2016 EFA annual meetingin Oslo, the 2016 AFFI Paris December meeting, and the 2017 CIB workshop on Dynamical Models in Finance. Theresearch leading to these results has received funding from the European Research Council under the EuropeanUnion’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 307465-POLYTE.†Swissquote Bank. Email: damien.ackerer@swissquote.ch‡EPFL and Swiss Finance Institute. Email: damir.filipovic@epfl.ch

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1 Introduction

We introduce a novel class of flexible and tractable reduced form models for the term structure ofcredit risk, the linear credit risk models. We directly specify the survival process of a firm, thatis, its conditional survival probability given the economic background information. Specifically, weassume a multivariate factor process with a linear drift and let the drift of the survival processbe linear in the factors. Prices of defaultable bonds and credit default swaps (CDS) are givenin closed form by linear-rational functions in the factors. By linearity, the same result holds forthe prices of CDSs on indices (CDISs). The implied default intensity is a linear-rational functionof the factors. In contrast, the price of a CDS in an affine default intensity model is a sum ofexponential-affine functions in the factors process and whose coefficients are given by the solutionsof nonlinear ordinary differential equations that are not in closed form, in general. In addition, thelinear credit risk models offer new tractable features such as a multi-name model with negativelycorrelated default intensity.

Within the linear framework we define the linear hypercube (LHC) model which is a single-namemodel. The factor process is diffusive with quadratic diffusion function so that it takes values in ahypercube whose edges’ length is given by the survival process. The quadratic diffusion functionis concave and bi-monotonic. This feature allows factors to virtually jump between low and highvalues. This facilitates the persistence and likelihood of term structure shifts. The factors’ volatilityparameters do not enter the bond and CDS pricing formulas, yet they impact the volatility of CDSspreads and thus affect CDS option prices. This may facilitate the joint calibration of credit spreadand option price time series. We discuss in detail the one-factor LHC model and compare it withthe one-factor affine default intensity model. We provide an identifiable canonical representationand the market price of risk specifications that preserve the linear drift of the factors.

We present a price approximation methodology for European style options on credit riskyunderlyings that exploits the compactness of the state space and the closed form of the conditionalmoments of the factor process. First, by the Stone–Weierstrass theorem, any continuous payofffunction on the compact state space can be approximated by a polynomial to any given level ofaccuracy. Second, the conditional expectation of any polynomial in the factors is a polynomial in theprevailing factor values. In consequence, the price of a CDS option can be uniformly approximatedby polynomials in the factors. This method also applies to the computation of credit valuationadjustments.

We build multi-name models by letting the survival processes be linear and polynomial com-binations of independent LHC models. Bonds and CDSs prices are still linear-rational but withrespect to an extended factor representation. These direct extensions can easily accommodate theinclusion of new factors and new firms. Stochastic short rate models with a similar specificationas the survival processes can be introduced while preserving the setup tractability. Simultaneousdefaults can be generated either by introducing a common jump process in the survival processesor a stochastic clock.

We perform an empirical and numerical analysis of the LHC model. Assuming a parsimoniouscascading drift structure, we fit two-factor and three-factor LHC models to the ten-year long timeseries of weekly CDS spreads on an investment grade and a high yield firm. The three-factor modelis able to capture the complex term structure dynamics remarkably well and performs significantlybetter than the two-factor model. We illustrate the numerical efficiency of the option pricing methodby approximating the prices of CDS options with different moneyness. Polynomials of relativelylow orders are sufficient to obtain accurate approximations for in-the-money options. Out-of-themoney options typically require a higher order. We derive the pricing formulas for CDIS optionsand tranches on a homogeneous portfolio to illustrate that their prices can also be approximated

2

with similar techniques. In general, the pricing of CDIS options and tranches requires manipulatingmultivariate polynomial bases of possibly large dimensions. In practice, computationally efficientmulti-name credit derivative pricing necessitates the use of special algorithms which are not studiedin this paper.

We now review some of the related literature. Our approach follows a standard doubly stochasticconstruction of default times as described in (Elliott, Jeanblanc, and Yor 2000) or (Bielecki andRutkowski 2002). The early contributions by (Lando 1998) and (Duffie and Singleton 1999) alreadymake use of affine factor processes. In contrast, the factor process in the LHC model is a strictly non-affine polynomial diffusion, whose general properties are studied in (Filipovic and Larsson 2016).The stochastic volatility models developed in (Hull and White 1987) and (Ackerer, Filipovic, andPulido 2018) are two other examples of non-affine polynomial models. Factors in the LHC modelshave a compact support and can exhibit jump-like dynamics similar to the multivariate Jacobiprocess introduced by (Gourieroux and Jasiak 2006). Our approach has some similarities with thelinearity generating process by (?) and the linear-rational models by (Filipovic, Larsson, and Trolle2017). These models also exploit the tractability of factor processes with linear drift but focus onthe pricing of non-defaultable assets. To our knowledge, we are the first to model directly thesurvival process of a firm with linear drift characteristics.

Options on CDS contracts are complex derivatives and intricate to price. The pricing andhedging of CDS options in a generic hazard process framework is discussed in (Bielecki, Jeanblanc,and Rutkowski 2006) and (Bielecki, Jeanblanc, Rutkowski, et al. 2008), and specialised to thesquare-root diffusion factor process in (Bielecki, Jeanblanc, and Rutkowski 2011). More recently(Brigo and El-Bachir 2010) developed a semi-analytical expression for CDS option prices in thecontext of a shifted square-root jump-diffusion default intensity model that was introduced in (Brigoand Alfonsi 2005). Another strand of the literature has focused on developing market models in thespirit of LIBOR market models. We refer the interested reader to (Schonbucher 2000), (Hull andWhite 2003), (Schonbucher 2004), (Jamshidian 2004), and (Brigo and Morini 2005). Black-Scholeslike formulas are then obtained for the prices of CDS options by assuming, for example, that theunderlying CDS spread follows a geometric Brownian motion under the survival measure. Althoughoffering more tractability, this approach makes it difficult, if not impossible, to consistently pricemultiple instruments exposed to the same source of credit risk. (Di Graziano and Rogers 2009)introduced a framework where they obtained closed form expressions similar to ours for CDS prices,but under the assumption that the firm default intensity is driven by a continuous-time finite-stateirreducible Markov chain.

Another important approach to default risk modeling is the use of subordinators to model thecumulative hazard process. It has been in particular shown that time-inhomogeneous models canreproduce well CDIS tranche prices. For more details on these models we refer to (Kokholm andNicolato 2010), (Sun, Mendoza-Arriaga, and Linetsky 2017), and references therein.

The simulation-based work by (Peng and Kou 2008) shows that a hazard-rate model withsystemic and idiosyncratic risk factors can fit both CDS and CDIS tranches, and therefore confirmsthat a bottom-up model with common risk factors can yield an accurate and fully consistent risk-management framework. A tractable alternative to price multi-name credit derivatives is to modelthe dependence between defaults with a copula function, as for example in (Li 2000), (Laurent andGregory 2005), and (Ackerer and Vatter 2017). However these models are by construction static,require repeated calibration, and in general become intractable when combined with stochasticsurvival processes as in (Schonbucher and Schubert 2001).

The idea of approximating option prices by power series can be traced back to (Jarrow andRudd 1982). However, most of the previous literature has focussed on approximating the transition

3

density function of the underlying process, see for example (Corrado and Su 1996) and (Filipovic,Mayerhofer, and Schneider 2013). In contrast, we approximate directly the payoff function by apolynomial.

The remainder of the paper is structured as follows. Section 2 presents the linear credit riskframework along with generic pricing formulas. Section 3 describes the single-name LHC model.The numerical and empirical analysis of the LHC model is in Section 4. Multi-name models aswell as models with stochastic interest rates are discussed in Section 5. Section 6 concludes. Theproofs are collected in the Appendix, as well as some additional results on market price of riskspecifications that preserve the linear drift of the factors, and on the two-dimensional Chebyshevinterpolation.

2 The linear framework

We introduce the linear credit risk model framework and derive closed form expressions for de-faultable bond prices and credit default swap spreads. We also discuss the pricing of credit indextranches, credit default swap options, and credit valuation adjustments.

2.1 Survival process specification

We fix a probability space (Ω,F ,Q) equipped with a right-continuous filtration F = (Ft)t≥0 repre-senting the economic background information, and where Q is the risk-neutral pricing measure. Weconsider N firms and let Si be the survival process of firm i. This is a right-continuous F-adaptedand non-increasing positive process with Si0 = 1. Let U1, . . . , UN be mutually independent standarduniform random variables that are independent from F∞. For each firm i, we define the randomdefault time

τi = inft ≥ 0 : Sit ≤ Ui,which is infinity if the set is empty. Let (Hit)t≥0 be the filtration generated by the indicator processwhich is one as long as firm i has not defaulted by time t and zero afterwards, H i

t = 1τi>t fort ≥ 0. The default time τi is a stopping time in the enlarged filtration (Ft ∨Hit)t≥0. It is F-doublystochastic in the sense that

Q[ τi > t | F∞ ] = Q[Sit > Ui | F∞ ] = Sit .

The filtration (Gt)t≥0 = (Ft ∨H1t ∨ · · · ∨ HNt )t≥0 contains all the information about the occurrence

of firm defaults, as well as the economic background information. Henceforward we omit the indexi of the firm and refer to any of the N firms as long as there is no ambiguity.

In a linear credit risk model the survival process of a firm is defined by

St = a>Yt, t ≥ 0, (1)

for some firm specific parameter a ∈ Rn+, and some common factor process (Y,X) taking values inRn+ × Rm with linear drift of the form

dYt = (c Yt + γ Xt)dt+ dMYt (2)

dXt = (b Yt + β Xt)dt+ dMXt (3)

for some c ∈ Rn×n, b ∈ Rm×n, γ ∈ Rn×m, β ∈ Rm×m, m-dimensional F-martingale MX , and n-dimensional F-martingaleMY . The process S being positive and non-increasing, we necessarily have

4

that its martingale component MS = a>MY is of finite variation and thus purely discontinuous,see (Jacod and Shiryaev 2013, Lemma I.4.14), and that −St− < ∆MS

t ≤ 0 for all t ≥ 0 because∆St = ∆MS

t . This observation motivates the decomposition of the factor process into a componentX and a component Y with finite variation. Although we do not specify further the dynamics ofthe factor process at the moment, it is important to emphasize that additional conditions shouldbe satisfied to ensure that S is a valid survival process.

Remark 2.1 In practice we will consider a componentwise non-increasing process Y with Y0 = 1.Survival processes can then easily be constructed by choosing any vector a ∈ Rn+ such that a>1 = 1.

The linear drift of the process (Y,X) implies that the Ft-conditional expectation of (Yu, Xu) islinear of the form

E[(Yu

Xu

) ∣∣∣ Ft] = eA(u−t)(YtXt

), t ≤ u, (4)

where the (m+ n)× (m+ n)-matrix A is defined by

A =

(c γb β

). (5)

Remark 2.2 If S is absolutely continuous, so that a>dMYt = 0 for all t ≥ 0, the corresponding

default intensity λ, which derives from the relation St = e−∫ t0 λsds, is linear-rational in (Y,X) of

the form

λt = −a>(c Yt + γ Xt)

St.

In this framework, the default times are correlated because the survival processes are driven bycommon factors. Simultaneous defaults are possible and may be caused by the martingale compo-nents of Y that forces the survival processes to jump downward at the same time. Additionally,and to the contrary of affine default intensity models, the linear credit risk framework allows fornegative correlation between default intensities as illustrated by the following stylized example.

Example 2.3 Consider the factor process (Y,X) taking values in R2+ × R defined by

dYt =ε

2

((−1 00 −1

)Yt +

(−11

)Xt

)dt

dXt = −κXtdt+ σ√

(e−εt −Xt)(e−εt +Xt)dWt

for some κ > ε > 0, σ > 0, X0 ∈ [−1, 1], and F-adapted univariate Brownian motion Wt. Theprocess X takes values in the interval [−e−εt, e−εt] at time t. Let N = 2 survival processes be definedby S1

t = Y1t and S2t = Y2t for all t ≥ 0 so that the implied default intensities of the two firms are

given by

λ1t =

ε

2

(1 +

Xt

Y1t

)and λ2

t =ε

2

(1− Xt

Y2t

), t ≥ 0.

This results in d〈λ1, λ2〉t ≤ 0 and d〈λ1, λ2〉t < 0 with positive probability, and λ1t , λ

2t ≤ ε. Moreover,

the default intensities, λ1 and λ2, both mean-revert towards ε/2. The proof of these statements isgiven in Appendix A.

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2.2 Defaultable bonds

We consider securities with notional amount equal to one and exposed to the credit risk of areference firm. We assume a constant risk-free interest rate equal to r so that the time-t price ofthe risk-free zero-coupon bond with maturity tM and notional amount one is given by e−r(tM−t).The following result gives a closed form expression for the price of a defaultable bond with constantrecovery rate at maturity.

Proposition 2.4 The time-t price of a defaultable zero-coupon bond with maturity tM and recoveryδ ∈ [0, 1] at maturity is

BM(t, tM ) = E[e−r(tM−t)

(1τ>tM + δ1τ≤tM

) ∣∣∣Gt]= (1− δ)BZ(t, tM ) + 1τ>tδ e−r(tM−t)

where BZ(t, tM ) = e−r(tM−t)E[1τ>tM | Gt] denotes the time-t price of a defaultable zero-couponbond with maturity tM and zero recovery. It is of the form

BZ(t, tM ) = 1τ>t1

a>YtψZ(t, tM )>

(YtXt

)(6)

where the vector ψZ(t, tM ) ∈ Rn+m is given by

ψZ(t, tM )> = e−r(tM−t)(a> 0>m

)eA(tM−t)

where the m-dimensional vector 0m contains only zeros.

The next result shows that the price of a defaultable bond paying a constant recovery rate atdefault can also be retrieved in closed form.

Proposition 2.5 The time-t price of a defaultable zero-coupon bond with maturity tM and recoveryδ ∈ [0, 1] at default is

BD(t, tM ) = E[e−r(tM−t)1τ>tM + δe−r(τ−t)1t<τ≤tM

∣∣∣Gt]= BZ(t, tM ) + δ CD(t, tM ),

where CD(t, tM ) = E[e−r(τ−t)1t<τ≤tM|Gt] denotes the time-t price of a contingent claim payingone at default if it occurs between dates t and tM . It is of the form

CD(t, tM ) = 1τ>t1

a>YtψD(t, tM )>

(YtXt

)(7)

where the vector ψD(t, tM ) ∈ Rn+m is given by

ψD(t, tM )> = −a>(c γ

) ∫ tM

teA∗(s−t)ds (8)

where A∗ = A− r Id.

The price of a security whose only cash flow is proportional to the default time is given inthe following corollary. It is used to compute the expected accrued interests at default for somecontingent securities such as CDS.

6

Corollary 2.6 The time-t price of a contingent claim paying τ at default if it occurs between datet and tM is of the form

CD∗(t, tM ) = E[τe−r(τ−t)1τ≤tM

∣∣∣Gt]= 1τ>t

1

a>YtψD∗(t, tM )>

(YtXt

) (9)

where the vector ψD∗(t, tM ) ∈ Rn+m is given by

ψD∗(t, tM )> = −a>(c γ

) ∫ tM

ts eA∗(s−t)ds. (10)

Note the presence of the factor s in the integrand on the right hand side of (10), which is absentin (8).

Remark 2.7 By setting r = 0 in (9), we obtain a closed form expression for E[τ1τ≤tM | Gt].This expression can be used to price a defaultable bond whose recovery value at maturity tM dependson the default time τ in a linear way,

BD0(t, t0, tM ) = BZ(t, tM ) + e−r(tM−t)E[(δ0

τ − t0tM − t0

+ δ1

)1τ≤tM

∣∣∣ Gt]for some parameters δ0, δ1 ≥ 0 such that δ0 + δ1 ≤ 1, and for some time t0 ≤ t.The following Lemma shows that pricing formulas (7)–(10) can further simplify with an additionalcondition.

Lemma 2.8 Assume that the matrix A∗ is invertible then we have the following closed form ex-pressions

ψD(t, tM )> = −a>(c γ

)A−1∗

(eA∗(tM−t) − Id

)ψD∗(t, tM )> = −a>

(c γ

) ((tM − t)A−1

∗ eA∗(tM−t)

+A−1∗ (Id t−A−1

∗ )(eA∗(tM−t) − Id))

where Id is the (n+m)-dimensional identity matrix.

This is a remarkable result since the prices of contingent cash flows become closed form expres-sions composed of basic matrix operations and are thus easily computed. Closed form formulas fordefaultables securities render the linear framework appealing for large scale applications, for exam-ple with a large number of firms and contracts, in comparison to standard affine default intensitymodels that in general require the use of additional numerical methods. For illustration, assumethat the survival process St is absolutely continuous so that it admits the default intensity λt as inRemark 2.2. Then CD(t, tM ) can be rewritten as

CD(t, tM ) = 1τ>t

∫ tM

te−r(u−t) E

[λue−

∫ ut λsds

∣∣∣Ft]du.With affine default intensity models the expectation to be integrated requires solving Riccati equa-tions, which have a closed form solution only when the default intensity is driven by a sum ofindependent univariate CIR processes. Numerical methods such as finite difference are usuallyemployed to compute the expectation with time-u cash flow for u ∈ [t, tM ]. The integral can thenonly be approximated by means of another numerical method such as quadrature, that necessitatessolving the corresponding ordinary differential equations at many different points u. For moredetails on affine default intensity models we refer to (?), (Filipovic 2009), and (Lando 2009).

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2.3 Credit default swaps

We derive closed form expressions for credit default swaps (CDS) on a single firm and multiplefirms. We conclude the section with a discussion of factors unspanned by bonds and CDS prices.

A single-name CDS is an insurance contract that pays at default the realized loss on a referencebond – the protection leg – in exchange of periodic payments that will stop after default – thepremium leg. We consider the following discrete tenor structure t ≤ t0 < t1 < · · · < tM and acontract offering default protection from date t0 to date tM . When t < t0 the contract is usuallycalled a knock out forward CDS and generates cash flows only if the firm has not defaulted bytime t0. We consider a CDS contract with notional amount equal to one. The time-t value of thepremium leg with spread k is given by k Vprem(t, t0, tM ) where

Vprem(t, t0, tM ) = Vcoup(t, t0, tM ) + Vai(t, t0, tM )

is the sum of the value of coupon payments before default

Vcoup(t, t0, tM ) =

M∑j=1

E[e−r(tj−t)(tj − tj−1)1tj<τ

∣∣∣Gt]and the value of the accrued coupon payment at the time of default

Vai(t, t0, tM ) =

M∑j=1

E[e−r(τ−t)(τ − tj−1)1tj−1<τ≤tj

∣∣∣Gt].The time-t value of the protection leg is

Vprot(t, t0, tM ) = (1− δ)E[e−r(τ−t)1t0<τ≤tM

∣∣∣Gt],where δ ∈ [0, 1] denotes the constant recovery rate at default. This specification of payments is inline with the ISDA model, see (?). The (forward) CDS spread CDS(t, t0, tM ) is the spread k thatmakes the premium leg and the protection leg equal in value at time t. That is,

CDS(t, t0, tM ) =Vprot(t, t0, tM )

Vprem(t, t0, tM ).

Proposition 2.9 The values of the protection and premium legs are given by

Vprot(t, t0, tM ) = 1τ>t1

Stψprot(t, t0, tM )>

(YtXt

)(11)

Vprem(t, t0, tM ) = 1τ>t1

Stψprem(t, t0, tM )>

(YtXt

)(12)

where the vectors ψprot(t, t0, tM ), ψprem(t, t0, tM ) ∈ Rn+m are given by

ψprot(t, t0, tM ) = (1− δ) (ψD(t, tM )− ψD(t, t0)) ,

ψprem(t, t0, tM ) =M∑j=1

(tj − tj−1)ψZ(t, tj) + ψD∗(t, tM )− ψD∗(t, t0)

+ tM−1ψD(t, tM )−M−1∑j=1

(tj − tj−1)ψD(t, tj)− t0ψD(t, t0).

8

As a consequence of Proposition 2.9, the CDS spread is given by a readily available linear-rationalexpression,

CDS(t, t0, tM ) = 1τ>t

ψprot(t, t0, tM )>(YtXt

)ψprem(t, t0, tM )>

(YtXt

) . (13)

This is a remarkably simple expression that allows us to see how the factors (Y,X) affect the CDSspread through the vectors ψprot(t, t0, tM ) and ψprem(t, t0, tM ). For comparison, in an affine defaultintensity model the two legs Vprot(t, t0, tM ) and Vprem(t, t0, tM ) are given as sums of exponential-affine terms that cannot be simplified further. In the following, we denote VCDS(t, t0, tM , k) thetime-t price of a CDS contract starting at time t0 with maturity tM and spread k,

VCDS(t, t0, tM , k) = 1τ>t(ψprot(t, t0, tM )− ψprem(t, t0, tM )

)>(YtXt

). (14)

A multi-name CDS, or credit default index swap (CDIS), is an insurance on a reference portfolioof N firms with equal weight which we assume to be 1/N so that the portfolio total notional amountis equal to one. The protection buyer pays a regular premium that is proportional to the currentnotional amount of the CDIS. Let δ ∈ [0, 1] be the recovery rate determined at inception. Upondefault of a firm the protection seller pays 1− δ to the protection buyer and the notional amountof the CDIS decreases by 1/N . These steps repeat until maturity or until all firms in the referenceportfolio have defaulted, whichever comes first.

Denote by Sit = a>i Yt the survival process of firm i as defined in (1). The CDIS spread simplifiesto a double linear-rational expression,

CDIS(t, t0, tM ) =

∑Ni=1 1τi>t (1/a>i Yt)ψ

iprot(t, t0, tM )>

(YtXt

)∑N

i=1 1τi>t (1/a>i Yt)ψiprem(t, t0, tM )>

(YtXt

)where ψiprot(t, t0, tM ) and ψiprem(t, t0, tM ) are defined as in Proposition 2.9 for each firm i.

Remark 2.10 (Unspanned Factors) The characteristics of the martingales MY and MX donot appear explicitly in the bond, CDS and CDIS pricing formulas. This leaves the freedom tospecify exogenous factors that feed into MY and MX . Such factors would be unspanned by theterm structures of defaultable bonds and CDS and give rise to unspanned stochastic volatility, asdescribed in (Filipovic, Larsson, and Trolle 2017). They provide additional flexibility for fittingtime series of bond prices and CDS spreads. These unspanned stochastic volatility factors affectthe distribution of the survival and factor processes and therefore can be recovered from the pricesof credit derivatives such as those discussed hereinafter.

2.4 CDIS tranche

A CDIS tranche is a partial insurance on the losses of a reference portfolio in the sense that onlylosses larger than the attachment point Ka and lower than the detachment point Kd are insured.We assume the same tenor structure and reference portfolio as for the CDIS contract, the protectionbuyer pays a periodic premium that is proportional to the current notional amount of the tranche,

Tt = (Kd −Ka − (Nt(1− δ)/N −Ka)+)+ (15)

9

where Nt =∑N

i=1 1τi≤t is the total number of firms which have defaulted in the reference portfolioat time t. The values of the protection leg and the premium leg at time t are respectively given by

Vprot(t, tM ,Ka,Kd) =

∫ tM

te−ru E[dTu | Gt], and

Vprem(t, tM ,Ka,Kd) =M∑j=1

e−rtj∫ tj

tj−1

(Kd −Ka − E[Tu | Gt]

)du.

(16)

The value of the tranche is then simply given by the difference of the cash flow values,

VT(t, tM ,Ka,Kd, k) = Vprot(t, tM ,Ka,Kd)− k Vprem(t, tM ,Ka,Kd) (17)

where k is the tranche spread. The following proposition shows that the (F∞ ∨ Gt)-conditionaldistribution of the number of defaults at time u > t, can be exactly retrieved in closed form byapplying the discrete Fourier transform as described in (Ackerer and Vatter 2017).

Proposition 2.11 The (F∞∨Gt)-conditional distribution of the number of defaults Nu is given by

Q[Nu = n | F∞ ∨ Gt] =1

N + 1

N∑j=0

ζnjN∏i=1

(ζj + (1− ζj)1τi>t

a>i Yu

a>i Yt

)(18)

for u > t, for any n = 0, . . . , N , and where ζ = exp(2iπ/(N + 1)) with the imaginary number i.

From (15) it follows immediately that the conditional expectation of Tu can be expressed as afunction of the conditional distribution of Nu. Assume for simplicity that Ka = na(1 − δ)/N andKd = nd(1 − δ)/N for some integers 0 ≤ na < nd ≤ N . Then the conditional expectation of Tu isgiven by

E[Tu | F∞ ∨ Gt] =

N−na∑j=1

(1− δ) min(j, nd − na)N

Q[Nu = na + j | F∞ ∨ Gt] (19)

for u > t.The tranche price (17) has therefore a closed form expression as long as the conditional proba-

bility Q[Nu = j | Gt] is available in closed form for all t ≤ u ≤ tM and j = 0, . . . , N . An example isgiven in Section 4.4 for a polynomial model.

2.5 CDS option and CDIS option

A CDS option with strike spread k is a European call option on the CDS contract exercisable onlyif the firm has not defaulted before the option maturity date t0. Its payoff at time t0 is given by

(VCDS(t0, t0, tM ))+ =1τ>t0

a>Yt0

(ψcds(t0, t0, tM , k)>

(Yt0Xt0

))+

withψcds(t, t0, tM , k) = ψprot(t, t0, tM )− k ψprem(t, t0, tM ). (20)

Denote by VCDSO(t, t0, tM , k) the price of the CDS option at time t,

VCDSO(t, t0, tM , k) = E[e−r(t0−t)

1τ>t0

a>Yt0

(ψcds(t0, t0, tM , k)>

(Yt0Xt0

))+ ∣∣∣ Gt]= 1τ>t

e−r(t0−t)

a>YtE[(ψcds(t0, t0, tM , k)>

(Yt0Xt0

))+ ∣∣∣ Ft]

10

where the second equality follows directly from Lemma A.1.A CDIS option gives the right at time t0 to enter a CDIS contract with strike spread k and

maturity tM on the firms in the reference portfolio which have not defaulted and, simultaneously,to receive the losses realized before the exercise date t0. Denote by VCDISO(t, t0, tM , k) the price ofthe CDIS option at time t ≤ t0,

VCDISO(t, t0, tM , k) =e−r(t0−t)

NE[( N∑

i=1

V iCDS(t0, t0, tM , k) + (1− δ)1τi≤t0

)+ ∣∣∣ Gt],where V i

CDS(t0, t0, tM , k) is defined as in (14) for firm i.

Proposition 2.12 The price of a CDIS option is given by

VCDISO(t, t0, tM , k) =∑

α∈0,1N

e−r(t0−t)

NE[(V∗(α, t0, tM , k))+q(α, t, t0)

∣∣Ft]with the conditional payoffs

V∗(α, t0, tM , k) =N∑i=1

αi

a>i Yt0ψicds(t0, t0, tM , k)>

(Yt0Xt0

)+ (1− δ)(1− αi)

and the conditional probabilities

q(α, t, t0) =N∏i=1

(a>i Yt0)αi(a>i (Yt − Yt0))1−αi

a>i Yt1τi>t + (1τi≤t)

1−αi

where α = (α1, . . . , αN ) and with the convention 00 = 0.

The time-t price of a CDS option, or of a CDIS option, is therefore given by the expectedvalue of a non-smooth continuous function in (Yt0 , Xt0) where t < t0. A methodology to price suchcontracts is presented in Section 3.2.

2.6 Credit valuation adjustment

The unilateral credit valuation adjustment (UCVA) of a position in a bilateral contract is thepresent value of losses resulting from its cancellation when the counterparty defaults.

Proposition 2.13 The time-t price of the UCVA with maturity tM and time-u net positive expo-sure f(u, Yu, Xu), for some continuous function f(u, y, x), is

UCVA(t, tM ) = E[e−r(τ−t)1t<τ≤tMf(τ, Yτ , Xτ )

∣∣∣Gt]=1τ>t

a>Yt

∫ tM

te−r(u−t)E

[f(u, Yu, Xu) a>(c Yu + γ Xu)

∣∣∣Ft]du.where τ is the counterparty default time.

Computing the UCVA therefore boils down to a numerical integration of European style optionprices. As is the case for CDS and CDIS options, these option prices can be uniformly approximatedas described in Section 3.2. We refer to (Brigo, Capponi, and Pallavicini 2014) for a thoroughanalysis of bilateral counterparty risk valuation in a doubly stochastic default framework.

11

3 The linear hypercube model

The linear hypercube (LHC) model is a single-name model, that is n = 1 so that S = Y . Thesurvival process is absolutely continuous, as in Remark 2.2, and the factor process X is diffusiveand takes values in a hypercube whose edges’ length is given by Yt, for all t ≥ 0. More formallythe state space of (Y,X) is given by

E =

(y, x) ∈ R1+m : y ∈ (0, 1] and x ∈ [0, y]m.

The dynamics of (Y,X) is

dYt = −γ>Xt dt

dXt = (bYt + βXt) dt+ Σ(Yt, Xt) dWt

(21)

for some γ ∈ Rm+ and some m-dimensional Brownian motion W , and where the volatility matrixΣ(y, x) is given by

Σ(y, x) = diag(σ1

√x1(y − x1), . . . , σm

√xm(y − xm)

)(22)

with volatility parameters σ1, . . . , σm ≥ 0.Let (Y,X) be an E-valued solution of (21). It is readily verified that Y is non-increasing and

that the parameter γ controls the speed at which it decreases,

0 ≤ γ>Xt ≤ γ>1Yt,

which implies

0 ≤ λt ≤ γ>1 and Yt ≥ Y0 e−γ>1 t > 0, for any t ≥ 0. (23)

Note that the default intensity upper bound γ>1 depends on γ, which is estimated from data.Therefore, a crucial step in the model validation procedure is to verify that the range of possibledefault intensities is sufficiently wide.

The following theorem gives conditions on the parameters such that the LHC model (21) is welldefined.

Theorem 3.1 Assume that, for all i = 1, . . . , m,

bi −∑j 6=i

β−ij ≥ 0, (24)

γi + βii + bi +∑j 6=i

(γj + βij)+ ≤ 0. (25)

Then for any initial law of (Y0, X0) with support in E there exists a unique in law E-valued solution(Y,X) of (21). It satisfies the boundary non-attainment, for any i = 1, . . . ,m,

(i) Xit > 0 for all t ≥ 0 if Xi0 > 0 and

bi −∑j 6=i

β−ij ≥σ2i

2, (26)

(ii) Xit < Yt for all t ≥ 0 if Xi0 < Y0 and

γi + βii + bi +∑j 6=i

(γj + βij)+ ≤ −σ

2i

2. (27)

12

The state space E is a regular (m + 1)-dimensional hyperpyramid. Figure 1 shows E whenm = 1 and illustrates the drift inward pointing conditions (24)–(25) at the boundaries of E.

In Section B we describe all possible market price of risk specifications under which the driftfunction of (Y,X) remains linear.

Remark 3.2 The volatility of Xi is maximal at the center of its support when Xi = Y/2 anddecreases to zero at its boundaries for Xi → 0 and Xi → Y . As a consequence, some factorsmay alternate visits to the lower part and upper part of their supports, and therefore may mimicregime-shifting behavior.

Remark 3.3 Define the normalized process Z = X/Y , then the dynamics of (Z, λ) is given by

dZt =(b+ (β + diag(γ>Zt))Zt

)dt+ Σ(1, Zt)dWt, (28)

dλt = γ>dZt. (29)

We derive closed form expressions for the stationary points of the drift of (Z, λ) in Sections 3.1and 4.1, and in Example 2.3.

3.1 One-factor LHC model

The default intensity of the one-factor LHC model, m = 1, has autonomous dynamics of the form

dλt = (λ2t + βλt + bγ)dt+ σ

√λt(γ − λt)dWt.

The diffusion function of λ is the same as the diffusion function of a Jacobi process taking valuesin the compact interval [0, γ]. However, the drift of λ includes a quadratic term that is neitherpresent in Jacobi nor in affine processes.1 Conditions (24)–(25) in Theorem 3.1 rewrite

b ≥ 0 and (γ + b+ β) ≤ 0.

That is, the drift of λ is nonnegative at λ = 0 and nonpositive at λ = γ. We can factorize the driftas follows,

λ2t + βλt + bγ = (λt − `1)(λt − `2),

for some roots 0 ≤ `1 ≤ γ ≤ `2. Hence λ drifts towards `1, as long as not λt = `2 = γ. Thecorresponding original parameters are given by β = −(`1 + `2) and bγ = `1`2, so that the drift ofthe factor X reads

βYt +BXt = (`1 + `2)

(`1`2

γ(`1 + `2)Yt −Xt

). (30)

As a sanity check we verify that the constant default intensity case, λt = γ for all t ≥ 0, isnested as a special case. This is equivalent to have X = Y , which can be obtained by specifying thedynamics dXt = −γXt dt for the factor process and the initial condition X0 = 1. This correspondsto the stationary points `1 = 0 and `2 = γ.

The dynamics of the standard one-factor affine model on R+ is

dλt = `2(`1 − λt)dt+ σ√λtdWt,

where `2 is the mean-reversion speed and `1 the mean-reversion level of λ. Figure 2 shows thedrift and diffusion functions of the default intensity for the one-factor LHC and affine models. The

1The Jacobi process has been used in (Delbaen and Shirakawa 2002) to model the short rate in which case therisk-free bond prices are given by weighted series of Jacobi polynomials in the short rate value.

13

drift function is affine in the affine model whereas it is quadratic in the LHC model. However, forreasonable parameters values, the drift functions look similar when the default intensity is smallerthan the mean-reversion level λ < `1. On the other hand when λ > `1, the force of drifting towards`1 is smaller and concave in the LHC model. The diffusion function is strictly increasing andconcave for the affine model whereas it has a concave semi-ellipse shape in the LHC model. Thediffusion functions have the same shape on [0, γ/2] but typically do not scale equivalently in theparameter σ. Note that the parameter γ can always be set sufficiently large so that the likelihoodof λ going above γ/2 is arbitrarily small.

3.2 Option price approximation

We saw in Sections 2.5 and 2.6 that the pricing of a CDS option, a CDIS option, or a UCVA boilsdown to computing a Ft-conditional expectation of the form

Φ(f ; t, tM ) = E[f(YtM , XtM ) | Ft]

for some continuous function f(y, x) on E. We now show how to approximate Φ(f ; t, tM ) inclosed form form by means of a polynomial approximation of f(y, x). The methodology presentedhereinafter applies to any linear credit risk model which has a compact state space E and for whichthe Ft-conditional moments of (YtM , XtM ) are computable.

To this end, we first recall how the Ft-conditional moments of (YtM , XtM ) for t ≤ tM can beobtained in closed form as described in (Filipovic and Larsson 2016). Denote by Poln(E) the setof polynomials p(y, x) on E of degree n or less. It is readily seen that the generator of (Y,X),

Gf(y, x) =(− γ>x (βy +Bx)>

)∇f(y, x) (31)

+1

2

m∑i=1

∂2f(y, x)

∂x2i

σ2i xi(y − xi), (32)

is polynomial in the sense that

GPoln(E) ⊂ Poln(E) for any n ∈ N.

Let Nn =(n+1+m

n

)denote the dimension of Poln(E) and fix a polynomial basis h1, . . . , hNn of

Poln(E). We define the function of (y, x)

Hn(y, x) = (h1(y, x), . . . , hNn(y, x))>

with values in RNn . There exists a unique matrix representation Gn of G |Poln(E) with respect tothis polynomial basis such that for any p ∈ Poln(E) we can write

Gp(y, x) = Hn(y, x)>Gn ~p

where ~p is the coordinate representation of p. This implies the moment formula

E[p(YtM , XtM ) | Ft] = Hn(Yt, Xt)>eGn(tM−t)~p (33)

for any t ≤ tM we have, see (Filipovic and Larsson 2016, Theorem (3.1)).

Remark 3.4 The choice for the basis Hn(y, x) of Poln(E) is arbitrary and one may simply considerthe monomial basis,

Hn(y, x) = 1, y, x1, . . . , xm, y2, yx1, x

21, . . . , x

nm

14

in which Gn is block-diagonal. There are efficient algorithms to compute the matrix exponentialeGn(tM−t), see for example (Higham 2008). Note that only the action of the matrix exponential isrequired, that is eGn(tM−t)~p for some p ∈ Poln(E), for which specific algorithms exist as well, seefor examples (Al-Mohy and Higham 2011) and (Sidje 1998) and references within.

Now let ε > 0. From the Stone-Weierstrass approximation theorem (Rudin 1974, Theorem 5.8)there exists a polynomial p ∈ Poln(E) for some n such that

sup(y,x)∈E

|f(y, x)− p(y, x)| ≤ ε. (34)

Combining (33) and (34) we obtain the desired approximation of Φ(f ; t, T ).

Theorem 3.5 Let p ∈ Poln(E) be as in (34). Then Φ(f ; t, tM ) is uniformly approximated by

supt≤tM

∥∥∥Φ(f ; t, tM )−Hn(Yt, Xt)>eGn(tM−t)~p

∥∥∥L∞≤ ε. (35)

The approximating polynomial p in (34) needs to be found case by case. We illustrate this forthe CDS option in Section 4.2 and for the CDIS option on an homogenous portfolio in Section 4.3.

Remark 3.6 Approximating the payoff function f(y, x) on a strict subset of the state space E issufficient to approximate an option price. Indeed, for any times t ≤ u ≤ s the process (Yu, Xu)t≤u≤stakes values in

(y, x) ∈ E : Yt ≥ y ≥ e−γ>1(s−t)Yt

⊂ E.

A polynomial approximation on a compact subset of E can be expected to be more precise and, asa result, to produce a more accurate price approximation. See Section 4.2 for an implementationexample.

4 Case studies

We show that the LHC model can reproduce complex term structure dynamics, that option pricescan be accurately approximated, and that the prices of derivatives on homogeneous portfolioscan similarly be computed. First, we fit a parsimonious LHC model specification to CDS dataand discuss the estimated parameters and factors. Then, we accurately approximate the price ofCDS options at different moneyness. Finally, for a homogeneous portfolio, we derive closed formexpressions for the payoff function of a CDIS option and for the tranche prices.

4.1 CDS calibration

We calibrate the LHC model to a high yield firm, Bombardier Inc., and also to an investment gradefirm, Walt Disney Co., in order to show that the model flexibly adjusts to different spread levelsand dynamics. We also present a fast filtering and calibration methodology which is specific toLHC models.

15

Data description. The empirical analysis is based on composite CDS spread data from Markitwhich are essentially averaged quotes provided by major market makers. The sample starts onJanuary 1th 2005 and ends on January 1th 2015. The data set contains 552 weekly observationssumming up to 3620 observed CDS spreads for each firm. At each date we include the availablespreads with the modified restructuring clause on contracts with maturities of 1, 2, 3, 4, 5, 7, and10 years.

Time series of the 1-year, 5-year, and 10-year CDS spreads are displayed in Figure 3, as wellas the relative changes on the 5-year versus 1-year CDS spread. The two term-structures of CDSspreads exhibit important fluctuations of their level, slope, and curvature. The time series can besplit into three time periods. The first period, before the subprime crisis, exhibits low spreads incontango and low volatility. The second period, during the subprime crisis, exhibits high volatilitywith skyrocketing spreads temporarily in backwardation. The crisis had a significantly largerimpact on the high yield firm for which the spreads have more than quadrupled. The third periodis characterized by a steep contango and a lot of volatility. Figure 3 also shows that CDS spreadchanges are strongly correlated across maturities. Summary statistics are reported in Table 1.

Model specification. The risk neutral dynamics of each survival process is given by the LHCmodel of Section 3 with two and three factors. We set γ = γ1e1, for some γ1 ≥ 0, and consider acascading structure of the form

dXit = κi(θiX(i+1)t −Xit) dt+ σi√Xit(Yt −Xit) dWit (36)

for i = 1, . . . , m− 1 and

dXmt = κm(θmYt −Xmt) dt+ σm√Xmt(Yt −Xmt) dWmt (37)

for some parameters κ, θ, σ ∈ Rm+ satisfying

θi ≤ 1− γ1

κi(38)

for i = 1, . . . , m. We have that βii = −κi, βi,i+i = κiθi, and βij = 0 otherwise, bm = κmθm andbi = 0 otherwise. It directly follows that

0 ≤ bi −∑j 6=i

β−ij = 1i=mκmθm = 1i=mβmm

and for i = 1, . . . , m

0 ≥ γi + βii + bi +∑j 6=i

(γj + βij)+ = γ1 − κi + κiθi

= γ1 + βii + 1i 6=mβi,i+1 + 1i=mbm.

This shows that the parameter conditions (24)-(25) are satisfied. Note that (24)-(25) boil downto standard linear parameter constraints when expressed in terms of β and b. They are thereforecompatible with efficient optimization algorithms.

This specification allows default intensity values to persistently be close to zero over extendedperiods of time. It also allows to work with a multidimensional model parsimoniously as the numberof free parameters is equal to 3m + 1 whereas it is equal to 3m + m2 for the generic LHC model.The default intensity is then proportional to the first factor and given by λ = γ1X1/Y .

16

We denote the two-factor and three-factor linear hypercube cascade models by LHCC(2) andLHCC(3), respectively. In addition, we estimate a three-factor model, denoted LHCC(3)∗, whereparameter γ1 is an exogenous fixed parameter. This parameter value is fixed so as to be abouttwice as large as the estimated γ1 from the LHCC(3) model. We estimate the constrained modelin order to determine whether the choice of the default intensity upper bound is critical for theempirical results.

We set the risk-free rate equal to the average 5-year risk-free yield over the sample, r = 2.52%.We make the usual assumption that the recovery rate is equal to δ = 40%. We also use Lemma 2.8to compute efficiently the CDS spreads, which is justified by the following result.

Lemma 4.1 Assume that r > 0, then the matrix A∗ = A − r Id with A as in (5) is invertible forthe cascade LHCC model defined in (36)–(37) and with γ = γ1e1.

Remark 4.2 The drift of the normalized process Z = X/Y admits the stationary points µt givenby the system of equations

µit = (−1)m−i+1m∏j=i

κjθjµ1tγ1 − κj

, i = 1, . . . ,m (39)

as shown in Appendix A. In fact, µ1t implies the values of µit for i = 2, . . . ,m. The stationarypoint of the drift of λ is given by γ1µ1t.

Filtering and calibration. We present an efficient methodology to filter the factors from theCDS spreads. We recall that the CDS spread CDS(t, t0, tM ) is the strike spread that renders theinitial values of the CDS contract equal to zero. We therefore obtain the affine equation

ψcds(t, t0, tM ,CDS(t, t0, tM ))>(

1Zt

)= 0 (40)

conditional on τ > t, and with the normalized process Z = X/Y ∈ [0, 1]m. Therefore, in theory wecould extract the value Zt from the observation of at least m spreads with different maturities. Thefactor value (St, Xt) at time t can in turn be inferred, for example, by applying the Euler schemeto compute the survival process value, for example, and then rescaling the pseudo factor Zt,

Yti = Yti−1 − γ>Xti−1∆t and Xti = YtiZti (41)

for all the observation dates ti, and with Yt0 = 1. In practice, there might not be a value Zt suchthat (40) is satisfied for all observed market spreads. Therefore, we consider all the observablespreads and minimize the following weighted mean squared error

minz

1

2

ni∑k=1

( ψcds(ti, ti, tkM ,CDS(ti, ti, t

kM ))>

(1z

)ψprem(ti, ti, tkM )>

(1

Zti−1

) )2

s.t. 0 ≤ zi ≤ 1, i = 1, . . . ,m

(42)

where t1M , . . . , tniM are the maturities of the ni observed spreads at date ti, and ti−1 is the previous

observation date. Dividing the CDS price error by an approximation of the CDS premium legvalue gives an accurate approximation of the CDS spread error when Zti ≈ Zti−1 . The above

17

minimization problem is a linearly constrained quadratic optimization problem which can be solvedvirtually instantaneously numerically.

For any parameter set we can extract the observable factor process at each date by recursivelysolving (42) and applying (41). With the parameters and the factor process values we can inturn compute the difference between the model and market CDS spreads. Therefore, we numeri-cally search the parameter set that minimizes the aggregated CDS spread root-mean-squared-error(RMSE) by using the gradient-free Nelder-Mead algorithm together with a penalty term to enforcethe parameter constraints and starting from several randomized initial parameter sets.

Note that we do not calibrate the volatility parameters σi for i = 1, . . . ,m since CDS spreadsdo not depend on the martingale components with linear credit risk models and since the factorprocess is observable directly from the CDS spreads. Furthermore, we only fit the risk-neutral driftparameters κ and θ implied by the CDS spreads. The total number of parameters for LHCC(2),LHCC(3), and LHCC(3)∗ model is therefore equal to 5, 7, and 6 respectively. Equipped with a fastfilter and a low dimensional parameter space, the calibration procedure is swift.

Remark 4.3 Alternatively one could estimate the parameters by performing a quasi-maximumlikelihood estimation or a more advanced generalized method of moments estimation. This can beimplemented in a straightforward manner with the LHC model if the market price of risk specifi-cation preserves the polynomial property of the factors as the real–world conditional moments of(Y,X) are then given in closed form, see Appendix B. The availability of conditional moments alsoenables direct usage of the Unscented Kalman Filter to recover the factor values at each date. How-ever this approach comes at the cost of more parameters and possibly more stringent conditions onthem, as well as unnecessary computational costs if we are only interested in market prices.

Parameters, fitted spreads, and factors. The fitted parameters are reported in Table 2. Animportant observation is that the parameter constraint in (38) is binding for each dimension in allthe fitted models. The calibrated parameter values are similar across the different specificationswhich is comforting, and the calibrated default intensity upper bounds appear large enough tocover the high spread values observed during the subprime crisis.

The fitted factors extracted from the calibration are used as input to compute the fitted spreads.With the fitted spreads we compute the fitting errors for each date and maturity. Not surprisinglythe more flexible specification LHCC(3) performs the best. Estimating the default intensity upperbound γ1 instead of setting an arbitrarily large value improves the calibration. Table 3 reportssummary statistics of the errors by maturity. The LHCC(3) model has the smallest RMSE for eachmaturity. In particular, its overall RMSE is half the one of the two-factor model. The LHCC(3)∗

model faces difficulties in reproducing long-term spreads as, for example, its RMSE is twice as largeas the one of the unconstrained LHCC(3) for the 10-year maturity spread for both firms. Figure 4displays the fitted spreads and the RMSE time series. Again, the LHCC(3) appears to have thesmallest level of errors over time. The two other models do not perform as good during the lowspreads period before the financial crisis, and during the recent volatile period. Overall, the fittedmodels appear to reproduce relatively well the observed CDS spread values.

Figure 5 shows the estimated factors. They are remarkably similar across the different specifica-tions. The default intensity explodes and the survival process decreases rapidly during the financialcrisis. The m-th factor controls the long term default intensity level. The second factors controlsthe medium term behavior of the term-structure of credit risk in the LHCC(3) and LHCC(3)∗

models. The LHCC(2) model requires an almost equal to zero default intensity to capture thesteep contango of the term structure at the end of the sample period, even lower than before thefinancial crisis. This seems counterfactual and illustrates the limitations of the LHCC(2) model

18

in capturing changing dynamics. The m-th factor visits the second half of its support [0, Yt] andappears to stabilize in this region for the three models.

4.2 CDS option pricing

We describe an accurate and efficient methodology to price CDS options that builds on the payoffapproximation approach presented in Section 3.2, and illustrate it with numerical examples. Themodel used for the numerical illustration is the one-factor LHC model from Section 3.1 with stylizedbut realistic parameters γ = 0.25, `1 = 0.05, `2 = 1, σ = 0.75, X0 = 0.2, and r = 0.

From Section 2.5, we know that the time-t CDS option price with strike spread k is of the form

VCDSO(t, t0, tM , k) = 1τ>tE[f(Z(t0, tM , k)) | Ft]

with the payoff function f(z) = e−r(t0−t)z+/Yt and where the random variable Z(t0, tM , k) is definedby

Z(t0, tM , k) = ψcds(t0, t0, tM , k)>(Yt0Xt0

)(43)

with ψcds(t0, t0, tM , k) as in (20). Furthermore, the random variable Z(t0, tM , k) takes values in theinterval [bmin, bmax] with the LHC model which is given by

bmin =

m+1∑i=1

min(0, ψcds(t0, t0, tM , k)i), and

bmax =m+1∑i=1

max(0, ψcds(t0, t0, tM , k)i).

We now show how to approximate the payoff function f with a polynomial by truncating its Fourier-Legendre series, and then how the conditional moments of Z(t0, tM , k) can be computed recursivelyfrom the conditional moments of (Yt0 , Xt0).

Let Len(x) denote the generalized Legendre polynomials taking values on the closed interval[bmin, bmax] and given by

Len(x) =

√1 + 2n

2σ2Len

(x− µσ

)where µ = (bmax + bmin)/2, σ = (bmax − bmin)/2, and the standard Legendre polynomials Len(x)on [−1, 1] are defined recursively by

Len+1(x) =2n+ 1

n+ 1xLen(x)− n

n+ 1Len−1(x)

with Le0 = 1 and Le1(x) = x. The generalized Legendre polynomials form a complete orthonormalsystem on [bmin, bmax] in the sense that the mean squared error of the Fourier-Legendre seriesapproximation f (n)(x) of any piecewise continuous function f(x), defined by

f (n)(x) =n∑k=0

fn Len(x), where fn =

∫ bmax

bmin

f(x)Len(x) dx, (44)

converges to zero,

limn→∞

∫ bmax

bmin

(f(x)− f (n)(x)

)2dx = 0.

19

The coefficients for the CDS option payoff are given in closed form,

fn = 1τ>te−r(t0−t)

Yt

∫ bmax

0z Len(z) dz,

since the integrands are polynomial functions. Note that a similar approach is followed in (Ackerer,Filipovic, and Pulido 2018) on the unbounded interval R with a Gaussian weight function.

The Ft-conditional moments of Z(t0, tM , k) can be computed recursively from the conditionalmoments of (Yt0 , Xt0). Let π : E 7→ 1, . . . , Nn be an enumeration of the set of exponents withtotal order less or equal to n, that is

E =α ∈ N1+m :

1+m∑i=1

αi ≤ n.

Define the polynomials

hπ(α)(s, x) = sα1

m∏i=1

xα1+i

i ,

which form a basis of Poln(E). Denote by 1 the (1 +m)-dimensional vector of ones and by ei the(1 +m)-dimensional vector whose i-th coordinate is equal to one and zero otherwise.

Lemma 4.4 For all n ≥ 2 we have

E[Z(t0, tM , k)n | Ft] =∑

α>1=n

cπ(α) E[hπ(α)(Yt0 , Xt0)

∣∣Ft]where the coefficients cπ(α) are recursively given by

cπ(α) =

1+m∑i=1

1αi−1≥0 cπ(α−ei) ψcds(t0, t0, tM , k)i.

We now report the main numerical findings. We take t0 = 1, tM = t0 + 5, and three referencestrike spreads k ∈ 250, 300, 350 basis points that represent in, at, and out of the money CDSoptions. The first row in Figure 6 shows the payoff approximation f (n)(z) in (44) for the polynomialorders n ∈ 1, 5, 30 and the strike spreads k ∈ 250, 300, 350. A more accurate approximation ofthe hockey stick payoff function is naturally obtained by increasing the order n, especially aroundthe kink. The width of the support [bmin, bmax] increases with the strike spread k, hence the uniformerror bound should be expected to be larger for out of the money options. This is confirmed by thesecond row of Figure 6 that shows the error bound (35) as a function of the approximation ordern for the Fourier-Legendre approach described above. It also displays the error bound when theCDS option payoff function is interpolated by means of Chebyshev polynomials, see Appendix C formore details. The error bound is approximated by taking the maximum distance between the payofffunction and the polynomial approximation on a regular grid of 104 points over [bmin, bmax]. Weremark that the error bound of the Chebyshev approach is oscillating around the error bound of theFourier-Legendre approach. This seems to be caused by variation of the polynomial approximationaccuracy around the payoff kink as the Chebyshev nodes change. Note that the error bound istypically non tight in practice, as illustrated in the following pricing application in which thepricing error is far lower than the error bound at least for n ≤ 20.

Figure 7 shows the price approximation as a function of the polynomial order, up to n = 30.The price approximations stabilize rapidly with the Fourier-Legendre approach so that a price

20

approximation using the first n = 10 moments appear to be accurate up to a basis point. Onthe other hand, the price approximations exhibit large oscillations with the Chebyshev approach.Figure 7 also shows that it takes a fraction of a second on a standard desktop to compute the priceapproximation. Note that almost all of the CPU time is spent on the computation of the momentsof Z(t0, tM , k).

We recall that the volatility parameter σ of the LHC model does not affect the CDS spreads, andcan therefore be used to improve the joint calibration of CDS and CDS options. We illustrate thisin the left panel of Figure 8 where the CDS option price is displayed as a function of the volatilityparameter for different strike spreads. As expected, the option price is an increasing function of thevolatility parameter. The right panel of Figure 8 also shows that X0 has an almost linear impacton the CDS option price.

Note that the dimension of the polynomial basis(

1+m+nn

)becomes a programming and compu-

tational challenge when both the expansion order n and the number of factors 1 +m are large. Forexample, for n = 20 and 1 + m = 2 the basis has dimension 231 whereas it has dimension 10’626when 1 + m = 4. In practice, we successfully implemented examples with 1 + m = 4 and n = 50on a standard desktop computer, in which case the basis dimension is 316’251.

4.3 CDIS option pricing

We discuss the approximation of the payoff function by means of Chebyshev polynomials for a CDISoption on a homogeneous portfolio. Let Nt =

∑Ni=0 1τi≤t denotes the number of firms which have

defaulted by time t. Consider a CDIS option on a homogeneous portfolio so that Sit = a>Yt for alli = 1, . . . , N . From Proposition 2.12 it follows that the time-t price of the CDIS option is given by

VCDISO(t, t0, tM , k) =e−r(t0−t)

N

N−Nt∑j=0

E[(V∗(j, t0, tm))+ q(j, t, t0)

∣∣Ft]with the conditional payoffs

V∗(j, t0, tm) =j

a>Yt0ψcds(t0, t0, tM , k)>

(Yt0Xt0

)+ (1− δ)(N − j)

and the conditional probabilities

q(j, t, t0) =

(N −Nt

j

)(a>Yt0)j(a>Yt − a>Yt0)N−Nt−j

(a>Yt)N−Nt(45)

with the notable difference that now the summation contains at most N + 1 terms because thedefaults are symmetric and thus interchangeable. Define the random variables

Y (t0) = a>Yt0 and X(t0, tM , k) = ψcds(t0, t0, tM , k)>(Yt0Xt0

).

The CDIS option price then rewrites

VCDISO(t, t0, tM , k) = E[f(Y (t0), X(t0, tM , k)) | Ft ∨Nt]

where the bivariate payoff function f(y, x) is given by

f(y, x) =e−r(t0−t)

N (a>Yt)N−Nt

((1− δ)N(a>Yt − y)N−Nt

+

N−Nt∑j=1

(N −Nt

j

)(j x+ y(1− δ)(N − j))+ yj−1(a>Yt − y)N−Nt−j

).

21

The Ft-conditional moments of (Y (t0), X(t0, tM , k)) can be computed recursively in a similar wayas in Lemma 4.4. The payoff function f(y, x) can be approximated using Chebyshev polynomialsand nodes, see Appendix C, or using its two-dimensional Fourier-Legendre series representation.

4.4 CDIS tranche pricing

As in Section 4.3, we consider a homogeneous portfolio so that Si = a>Y for all i = 1, . . . , N . Inthis case, a simpler expression for (18) can be derived,

Q[Nu = j | F∞ ∨ Gt] = Q[N −Nu = N − j | F∞ ∨ Gt]= q(N − j, t, u)

(46)

for u > t and j = Nt, . . . , N , and where q(N − j, t, u) is defined as in (45). We fix the attachmentpoint to Ka = na(1− δ)/N and the detachment point to Kd = nd(1− δ)/N , for some integers0 ≤ na < nd ≤ N . Assuming for simplicity that Nt ≤ na, then from (19) and (46) we obtain that

E[Tu | F∞ ∨ Gt] =

N∑j=na+1

(1− δ) min(j − na, nd − na)N

q(N − j, t, u)

and, by differentiating with respect to u,

dE[Tu | F∞ ∨ Gt]du

=N∑

j=na+1

(1− δ) min(j − na, nd − na)N

×(N −Nt

N − j

)(a>Yu)N−j−1(a>Yt − a>Yu)j−Nt−1

(a>Yt)N−Nt

×((N − j)a>Yt − (N −Nt)a

>Yu)a>(c Yu + γ Xu)

for any u > t. The protection and premium legs in (16) can thus, in principle, be computed inclosed form using the moments formula (33).

5 Extensions

We present several model extensions offering additional features. We first construct multi-namemodels, then include stochastic interest rates possibly correlated with credit spreads, and concludeby discussing jumps and stochastic clocks to generate simultaneous defaults.

5.1 Multi-name models

We build upon the LHC model to construct multi-name models with correlated default intensitiesand which can easily accommodate the inclusion of new factors and firms. This approach can beapplied to other linear credit risk models, as long as they belong to the class of polynomial models.We consider n independent LHC processes

(Y 1, X1), . . . , (Y n, Xn) (47)

where each (Y j , Xj) is defined as in (21)–(22). We defined the stacked processes Y = (Y 1, . . . , Y n)with Y0 = 1 and X = (X1, . . . , Xn) with X0 ∈ [0, 1]m where m =

∑nj=1mj . We denote E the state

space of (Y,X).

22

Let h = (h1, . . . , hn) be the Rn+-valued process whose j-th component is given by

hjt =γj>Xjt

Y jt

, t ≥ 0 (48)

where the vector γj ∈ Rmj is the drift parameter of Y j , see (21).

Linear Construction. The survival process of the firm i = 1, . . . , N can be defined as in (1),Si = a>i Y , for some vector ai ∈ Rn+ satisfying a>1 = 1. The corresponding default intensity λi

of firm i is for all t ≥ 0 given by a weighted sum of h, that is λit = wit>ht with stochastic weights

wijt = aijYjt /S

it > 0 satisfying

∑dj=1w

ijt = 1.

Polynomial Construction. Fix a degree d and define the survival process of each firm i =1, . . . , N by Sit = pi(Yt) for all t ≥ 0, for some polynomial pi(y) ∈ Pold([0, 1]n) which is componen-twise non-increasing and positive on [0, 1]n, and such that pi(1) = 1. Let Hd(y, x) be a polynomialbasis of Pold(E) stacked in a row vector and of the form Hd(y, x) = (Hd(y), H∗d(y, x)) where Hd(y)is itself a polynomial basis of Pold([0, 1]n). The survival process of firm i then rewrites Si = a>i Ywith the finite variation process Y = Hd(Y ), the factor process X = H∗d(Y,X) and where thevector ai is given by the equation pi(y) = Hd(y) ai. It follows from the polynomial property thatthe process (Y,X ) has a linear drift as in (2)–(3), see (Filipovic and Larsson 2017, Theorem 4.3).The specific values for the drift of (Y,X ) depend on the choice of the polynomial basis Hd(y, x).

Example 5.1 Take p(y) = yα =∏ni=1 y

αii for some α ∈ Nn, then the implied default intensity is

a weighted sum λt = α>ht with ht as defined in (48). The weights are constant as opposed to thestochastic weights in the linear construction.

Remark 5.2 The dimension of Hd(y, x) is(d+n+m

d

)and may be large depending on the values

of m + n and d. However, given that the pairs (Y it , X

it) in (47) are independent, the conditional

expectation of a monomial in (Yu, Xu) rewrites

E[ n∏i=1

(Y iu)αi(Xi

u)βi∣∣∣ Ft] =

n∏i=1

E[(Y iu)αi(Xi

u)βi∣∣∣ Ft], u > t,

for some αi ∈ N and βi ∈ Nmj for all i = 1, . . . , n. Hence, to compute bonds and CDSs prices weonly need to consider n independent polynomial bases of total dimension equal to

∑ni=1

(d+1+mi

d

).

5.2 Stochastic interest rates

We include stochastic interest rates possibly correlated with credit spreads. We denote the discountprocess Dt = exp(−

∫ t0 rsds), for t ≥ 0, where rs is the short rate value at time s. We specify that

D = a>r Y for some vector ar ∈ Rn. This is similar to the specification of the survival process of afirm, but we do not require that D is non-increasing. That is, we allow for negative interest rates.We follow Section 5.1 and let H2(y, x) be a polynomial basis of Pol2(E) which defines a new linearcredit risk model (Y,X ) = (H2(Y ), H∗2 (Y,X)) whose linear drift is given by a matrix A as in (5).

Proposition 5.3 The pricing formulas (6), (7), and (9) also apply with (Yt,Xt) in place of(Yt, Xt), with r = 0, by using the vector

ψZ(t, tM )> =(a>Z 0

)eA(tM−t)

23

where the vector aZ is given by H2(y)>aZ = (a>r y)(a>y), and the vectors

ψD(t, tM )> = a>D

∫ tM

teA(s−t)ds,

ψD∗(t, tM )> = a>D

∫ tM

ts eA(s−t)ds,

where the vector aD is given by H2(y, x) aD = (a>r y)(−a>(cy γx)).

In practice it can be sufficient to consider a basis strictly smaller than H2(y, x), as the followingexample suggests.

Example 5.4 Consider two independent LHC processes (Y j , Xj) with mj = 1 for j ∈ 1, 2, andconsider the following linear credit risk model with stochastic interest rate given by,

Dt = Y 1t and St = ν Y 1

t + (1− ν)Y 2t , for all t ≥ 0,

for some parameter ν ∈ (0, 1). The calculation of bond and CDS prices only requires the subbases

H0(y, x) =(y2

1 y1 y2

), H1(y, x) =

(y1x1 y1x2 x1y2 x2

1 x1x2

),

whose total dimension is dim((H0(y, x), H1(y, x))) = 7 < dim(Pol2(E)) = 15. The drift term of theprocess (H0(Y,X), H1(Y,X)) is

A =

0 0 −2γ1 0 0 0 00 0 0 −γ2 −γ1 0 0b1 0 β1 0 0 −γ1 00 b2 0 β2 0 0 −γ1

0 b1 0 0 β1 0 0σ2

1 0 2b1 − σ21 0 0 2β1 0

0 0 0 b1 b2 0 β1 + β2

where the subscripts indicate the LHC model identity. The pricing vectors in this basis are

aZ =(ν 1− ν

)and aD =

(0 0 −ν γ1 −(1− ν) γ2 0 0 0

).

5.3 Jumps and simultaneous defaults

There are two ways to include jumps in the survival process dynamics that may result in thesimultaneous default of several firms. The first is to let the martingale part of Y be driven by ajump process so that multiple survival processes may jump at the same time. The second is to lettime run with a stochastic clock leaping forward hence producing synchronous jumps in the factorsand the survival processes.

The survival process remains defined as in (1) but the factors are extensions of the LHC processin what follows. For simplicity, we discuss a unique pair (Y,X) as in (21) whose parametersγ, β,B satisfy (24)–(25). Let Z be a nondecreasing Levy process with Levy measure νZ(dζ) anddrift bZ ≥ 0 that is independent from the Brownian motion W and the uniform random variablesU1, . . . , UN .

24

Jump-Diffusion Model. Assume that ∆Zt ≤ 1 for all t ≥ 0. We define the dynamics of theLHC model with jumps as follows

d

(YtXt

)=

(−c −γ> − δ>E[Z1]b β − diag(ν)E[Z1]

)(Yt−Xt−

)dt+

(0

Σ(Yt−, Xt−)

)dWt

−(c Yt− + δ>Xt−

diag(ν)Xt−

)dNt

with the martingale N given by Nt = Zt − E[Z1]t for t ≥ 0, for some c > 0, δ ∈ Rm+ , and ν ∈ Rm+such that

c+ δ>1 < 1, c+ δ>1 ≤ νi ≤ 1, i = 1, . . . ,m (49)

and νi < 1 if (26) applies, i = 1, . . . ,m (50)

Conditions (49)–(50) ensure that the process always jumps inside its state space. Note that thesame process Z can affect the dynamics of multiple LHC processes (Y i, Xi).

Stochastic Clock. We consider the time-changed process (Yt, Xt)t≥0 = (YZt , XZt)t≥0 that willdirectly feed into (1) in place of (Yt, Xt) and whose factor dynamics is given by(

dYtdXt

)= A

(YtXt

)dt+

(dM Y

t

dM Xt

)

where the (m+ n)× (m+ n)-matrix A is now given by

A = bZ A+

∫ ∞0

(eAζ − Id)νZ(dζ) (51)

with the matrix A as in Equation (5), see (Sato 1999, Chapter 6) and (Filipovic and Larsson 2017,Theorem 6.1). The time-changed LHC model remains a linear credit risk model. The backgroundfiltration F is now the natural filtration of the process (YZ , XZ). Denote Ψ(·) the Laplace exponentof Z defined by E[exp(−uZt)] = exp(−tΨ(u)). The following Proposition shows that the matrix Amay be computed in closed form2.

Proposition 5.5 Assume that A = UDU−1 where U is a unitary matrix and D is a diagonalmatrix with nonpositive entries, then A = −UΨ(−D)U−1.

In some cases, the expression for A simplifies and does not require factoring the matrix A as shownin the following example.

Example 5.6 Let Z be a Gamma process such that νZ(dζ) = γZζ−1e−λZζdζ for some constants

λZ , γZ > 0 and bZ = 0. If the eigenvalues of the matrix A have nonpositive real parts, the drift ofthe time changed process (YZ , XZ) is then equal to

A = −γZ log(

Id−Aλ−1Z

)(52)

as shown in Appendix A.

2We thank an anonymous referee for suggesting this result.

25

Survival processes built from independent LHC models can be time changed with the samestochastic clock Z in order to generate simultaneous defaults and thus default correlation. Notethat the idea of using time change to generate simultaneous jumps in the cumulative hazard orthe survival processes is not new, see for example (Mendoza-Arriaga and Linetsky 2016) for anearlier contribution where a multi-name unified credit-equity model with simultaneous defaults isdeveloped.

Remark 5.7 One could use the Additive subordinators presented in (Li, Li, and Mendoza-Arriaga2016) in order to increase the model’s flexibility. These subordinators are time-dependent and maytherefore help to better fit term structures at the cost of introducing additional parameters. In thiscase, the drift of the factor process (Y , X) remains linear but the matrix A in (51) may then betime-dependent and may not have a closed form representation which would in turn lead to highercomputational costs.

6 Conclusion

The class of linear credit risk models is rich and offers new modeling possibilities. The survivalprocess and its drift are linear in the factor process whose drift is also linear. Consequently, theprices of defaultable bonds, credit default swaps (CDSs), and credit default index swaps (CDISs)become linear-rational expressions in the factors. We introduce and study the single-name lin-ear hypercube (LHC) model which consists of a diffusive factor process with a quadratic diffusionfunction and taking values in a compact state space. These features are employed to develop anefficient European option pricing methodology. By building upon the LHC model, we constructparsimonious and versatile multi-name models. The setup can accommodate stochastic interestrates correlated with credit spreads by constructing the discount process similarly as a survivalprocess. Jumps in the factor dynamics as well as stochastic clocks can be used to generate simulta-neous defaults. An empirical analysis shows that the LHC model can reproduce complex CDS termstructure dynamics. We numerically verify that CDS option prices at different moneyness can beaccurately approximated for the LHC model. We also show that CDIS option prices and trancheprices on a homogeneous portfolio can be approximated with the same approach. Future researchdirections include the development of efficient algorithms to price multi-name credit derivatives,and the joint empirical study of single-name and multi-name credit contracts.

A Proofs

This Appendix contains the proofs of all theorems and propositions in the main text.

Proof of (4)

This follows as in (Filipovic, Larsson, and Trolle 2017, Lemma 3).

Proof of Example 2.3

The autonomous process X admits a solution taking values in [−e−εt, e−εt] at time t with ε > 0and X0 ∈ [−1, 1] if and only if κ > ε, see (Filipovic and Larsson 2016, Theorem 5.1). The twocoordinates of Y are lower bounded by X. Indeed for i = 1, 2 we have

dYit = − ε2

(Yit ±Xt)dt ≥ −ε

2(Yit + e−εt)dt

26

The solution of dZt = −(ε/2)(Zt + e−εt)dt with Z0 = 1 is given by Zt = e−εt, t ≥ 0, which provesthat Yit ≥ Zt ≥ |Xt| for i = 1, 2. Finally, by applying Ito’s lemma we obtain

d〈λ1, λ2〉t = −ε2

4

σ2(e−εt −Xt)(e−εt +Xt)

Y1tY2t, t ≥ 0,

which is negative with positive probability. The dynamics of λi is given by,

dλit = (ε2/4)(± (1− 2κ/ε)(Xt/Yit) + (Xt/Yit)

2)dt± dMit

=((ε/2)(1− 2κ/ε)(λit − ε/2) + (λit − ε/2)2

)dt± dMit

where dMit = ε σ/(2Yit)√

(e−εt −Xt)(e−εt +Xt)dWt, and κ > ε. The quadratic drift of λi has twopositive roots, κ and ε/2, is positive at zero, and is negative at ε. Since κ > ε, this shows that λi

mean reverts towards ε/2 for i = 1, 2.

Proof of Proposition 2.4

Proposition 2.4 is an immediate consequence of (4) and the following lemma.

Lemma A.1 Let Y be a nonnegative F∞-measurable random variable. For any time t ≤ tM <∞,

E[1τ>tMY

∣∣Gt] = 1τ>t1

StE[StMY | Ft].

Note that tM <∞ is essential unless we assume that S∞ = 0.

Lemma A.1 follows from (Bielecki and Rutkowski 2002, Corollary 5.1.1). For the convenienceof the reader we provide here a sketch of its proof. As in (Bielecki and Rutkowski 2002, Lemma5.1.2) one can show that, for any nonnegative random variable Z, we have

E[1τ>tZ

∣∣Ht ∨ Ft] = 1τ>t1

StE[1τ>tZ

∣∣Ft].Setting Z = 1τ>tMY we can now derive

E[1τ>tMY

∣∣Gt] = E[1τ>tY 1τ>tM

∣∣Gt] = 1τ>t1

StE[1τ>tMY

∣∣Ft]= 1τ>t

1

StE[E[1τ>tM

∣∣F∞]Y ∣∣Ft]= 1τ>t

1

StE[StMY | Ft].

Proof of Proposition 2.5

The subsequent proofs build on the following lemma that follows from (Bielecki and Rutkowski2002, Proposition 5.1.1).

Lemma A.2 Let Z be a bounded F-predictable process. For any t ≤ tM <∞,

E[1t<τ≤tMZτ

∣∣Gt] = 1t<τ1

St

∫(t,tM ]

E[−ZudSu | Ft].

Note that tM <∞ is essential unless we assume that S∞ = 0.

27

We can now proceed to the proof of Proposition 2.5. The value of the contingent cash flow isgiven by the expression

CD(t, tM ) = E[e−r(τ−t)1t≤τ≤tM

∣∣∣Gt]By applying Lemma A.2 we get

CD(t, tM ) =1τ>t

St

∫ tM

tE[−e−r(s−t)dSs

∣∣∣Ft]=1τ>t

St

∫ tM

te−r(s−t)E

[−a>(cYs + γXs)

∣∣∣Ft]ds=1τ>t

St

∫ tM

te−r(s−t) − a>

(c γ

)eA(s−t)

(YtXt

)ds

where the second equality comes from the fact that∫ t

0 e−ru dMSu is a martingale. The third equality

follows from (4).

Proof of Corollary 2.6

The value of this contingent bond is given by

CD∗(t, tM ) = E[τ e−r(τ−t)1t<τ≤tM

∣∣∣Gt] =1τ>t

St

∫ tM

tE[−s e−r(s−t)dSs

∣∣∣Ft]and the result follows as in the proof of Proposition 2.5.

Proof of Lemma 2.8

Observe that for any matrix A and real r we have ereA = ediag(r)+A, and that the matrix exponentialintegration can be computed closed form as follows∫ u

0eAsds =

∫ u

0(I +As+A2 s

2

2+ . . . )ds = Iu+A

u2

2+A2u

3

6+ . . .

= A−1(eAu − I).

By change of variable u = s− t we obtain∫ tM

tseA∗(s−t)ds =

∫ tM−t

0ueA∗udu+ t

∫ tM−t

0eA∗udu,

where the second term on the RHS is given in Lemma 2.5. The first term can be derived usingintegration by parts∫ tM−t

0ueA∗udu = (tM − t)A−1

∗ eA∗(tM−t) −A−1∗ A−1

∗ (eA∗(tM−t) − I).

Proof of Proposition 2.9

The calculation of the protection leg and the coupon part, V iprot(t, t0, tM ) and V i

coup(t, t0, tM ) re-spectively, follows from Propositions 2.4 and 2.5. The accrued interest V i

ai(t, t0, tM ) is given by thesum of contingent cash flows and of weighted zero-recovery coupon bonds, and thus its calculation

28

follows from Propositions 2.5 and 2.6. The series of contingent cash flow is in fact equal to a singlecontingent payment paying τ at default,

CD∗(t, tM ) =

M∑j=1

E[τ e−r(τ−t)1tj−1<τ≤tj

∣∣∣Gt]= E

[τ e−r(τ−t)1t<τ≤tM

∣∣∣Gt].Using the identity 1tj−1<τ≤tj = 1τ>tj−1−1τ>tj we obtain that the second term of V i

ai(t, t0, tM )is given by

−M∑j=1

E[e−r(τ−t)tj−11tj−1<τ≤tj

∣∣∣Gt] =M∑j=1

tj−1 (CD(t, tj)− CD(t, tj−1))

= tM−1CD(t, tM )− T0CD(t, t0)−M−1∑j=1

(tj − tj−1)CD(t, tj).

Proof of Proposition 2.11

The conditional characteristic function of Nu is given by

φ(t, ξ) = E[

exp(iξNu

) ∣∣∣ F∞ ∨ Gt] = E[

exp(iξ

N∑i=1

1τi≤u) ∣∣∣ F∞ ∨ Gt]

= E[ N∏i=1

(1τi>u + eiξ(1− 1τi>u)

) ∣∣∣ F∞ ∨ Gt]=

N∏i=1

(1τi>tSit

(Siu + eiξ(Sit − Siu)) + 1τi≤teiξ)

=N∏i=1

(eiξ + 1τi>t(1− eiξ)

SiuSit

)where the first equality in the third line follows from (Bielecki and Rutkowski 2002, Lemma 9.1.3),which gives the expression

E[1τ1>t0,..., τN>t0 | Ft0 ∨ Gt

]=

N∏i=1

1τi>tSit0Sit. (53)

The expression (18) then directly follows by applying the discrete Fourier transform, see (Ackererand Vatter 2017, Section 3) for more details.

Proof of Proposition 2.12

The payoff at time t0 of the CDIS option can always be decomposed into 2N terms by conditioningon all the possible default events

q(α) =N∏i=1

((1τi>t0)

αi + (1τi≤t0)1−αi

)(54)

29

for α ∈ 0, 1N , and with the convention 00 = 0, so that the payoff function rewrites

( N∑i=1

1τi>t0

Sit0ψicds(t0, t0, tM , k)>

(Yt0Xt0

)+ (1− δ)1τi≤t0

)+

=∑

α∈0,1N

( N∑i=1

αiSit0

ψicds(t0, t0, tM , k)>(Yt0Xt0

)+ (1− δ)(1− αi)

)+q(α).

We can apply (Bielecki and Rutkowski 2002, Lemma 9.1.3) to compute the probability (53) so thatby writing (54) as a linear combination of indicator functions we obtain

q(α, t, t0) = E [q(α) | Ft0 ∨ Gt]

=

N∏i=1

((Sit0)αi(Sit − Sit0)1−αi

Sit1τi>t + (1τi≤t)

1−αi)

which completes the proof.

Proof of Theorem 3.1

We define the bounded continuous map (Y,X ) : R1+m → R1+m by

Y(y, x) = y+ ∧ 1, Xi(y, x) = x+i ∧ y+ ∧ 1, i = 1, . . . ,m,

such that (Y,X )(y, x) = (y, x) on E. In a similar vein, extend the dispersion matrix Σ(y, x) to abounded continuous mapping Σ((Y,X )(y, x)) on R1+m. The stochastic differential equation (21)then extends to R1+m by

dYt = −γ>X (Yt, Xt) dt

dXt = (bY(Yt) + βX (Yt, Xt)) dt+ Σ ((Y,X )(Yt, Xt)) dWt.(55)

Since drift and dispersion of (55) are bounded and continuous on R1+m, there exists a weak solution(Y,X) of (55) for any initial law of (Y0, X0) with support in E, see (Karatzas and Shreve 1991,Theorem V.4.22).

We now show that any weak solution (Y,X) of (55) with (Y0, X0) ∈ E stays in E,

(Yt, Xt) ∈ E for all t ≥ 0. (56)

To this end, for i = 1, . . . ,m, note that

Σii ((Y,X )(y, x)) = 0 for all (y, x) with xi ≤ 0 or xi ≥ y. (57)

Conditon (24) implies that

(bY(y) + βX (y, x))i ≥ 0 for all (y, x) with xi ≤ 0. (58)

For δ, ε > 0 we define

τδ,ε = inf t ≥ 0 | Xit ≤ −ε and −ε < Xis < 0 for all s ∈ [t− δ, t) .

Then on τδ,ε <∞ we have, in view of (57) and (58),

0 > Xiτδ,ε −Xiτδ,ε−δ =

∫ τδ,ε

τδ,ε−δ(bY(Yu) + βX (Yu, Xu))i du ≥ 0,

30

which is absurd. Hence τδ,ε = ∞ a.s. and therefore Xit ≥ 0 for all t ≥ 0. Similarly, conditon (25)implies that

−γ>X (y, x)− (bY(y) + βX (y, x))i ≥ 0 for all (y, x) with xi ≥ y. (59)

Using the same argument as above for Yt − Xit in lieu of Xit, and (59) in lieu of (58), we seethat Yt − Xit ≥ 0 for all t ≥ 0. Note that 0 ≤ γ>X (y, x) ≤ γ>1y+ for all (y, x), and thus

1 ≥ Yt ≥ e−γ>1t > 0 for all t ≥ 0. This proves (56) and thus the existence of an E-valued solution

of (21).Uniqueness in law of the E-valued solution (Y,X) of (21) follows from (Filipovic and Larsson

2016, Theorem 4.2) and the fact that E is relatively compact.The boundary non-attainment conditions (26)–(27) follow from (Filipovic and Larsson 2016,

Theorem 5.7(i) and (ii)) for the polynomials p(y, x) = xi and y − xi, for i = 1, . . . ,m.

Proof of Lemma 4.1

The matrix A∗ in the LHCC model is given by

A∗ =

−r −γ1 0 00 −(κ1 + r) κ1θ1 0 · · ·...

. . .

θm 0 −(κm + r)

and its determinant is therefore equal to

|A∗| = −r

∣∣∣∣∣∣∣−(κ1 + r) κ1θ1 0 · · ·

.... . .

0 0 −(κm + r)

∣∣∣∣∣∣∣+ (−1)m

∣∣∣∣∣∣∣∣∣−γ1 0 0

−(κ1 + r) κ1θ1 0 · · ·...

. . .

0 −(κm + r) κmθm

∣∣∣∣∣∣∣∣∣ .With r > 0, the first element on the right hand side is nonzero with sign equal to (−1)1+m and thesecond element also has a sign equal to (−1)1+m. This is because the determinant of a triangularmatrix is equal to the product of its diagonal elements. As a result, the determinant of A∗ isnonzero which concludes the proof.

Proof of Equation (39)

For i = 1, . . . ,m we have that d(1/Yt) = γ1Z1t/Yt for all t ≥ 0. The dynamics of Z is thus given by

dZit = (κiθiZ(i+1)t − κiZit + γ1Z1tZit)dt+ σi√Zit(1− Zit) dWit,

for i = 1, . . . ,m− 1, and by

dZmt = (κmθm − κmZmt + γ1Z1tZmt)dt+ σm√Zmt(1− Zmt) dWmt

Fixing Z1t = µ1t and solving for the value of Zmt which cancels its drift we obtain

µmt =−κmθm

µ1tγ1 − κm,

and solving recursively for i = m− 1, . . . , 1 gives (39).

31

Proof of Lemma 4.4

We n-th power of Z(t0, tM , k) rewrites

Z(t0, tM , k)n =(ψcds(t0, t0, tM , k)>

(Yt0Xt0

))n= ψcds(t0, t0, tM , k)>

(Yt0Xt0

) ∑α>1=n−1

cπ(α) hπ(α)(Yt0 , Xt0)

=1+m∑i=1

∑α>1=n−1

cπ(α)ψcds(t0, t0, tM , k)i hπ(α+ei)(Yt0 , Xt0)

which is a polynomial containing all and only polynomials of degree n, the lemma follows byrearranging the terms.

Proof of Proposition 5.3

The time-t price of the zero-coupon zero-recovery bond is now given by

BZ(t, tM ) = E[DtM

Dt1τ>tM

∣∣∣ Gt] =1τ>t

DtStE[DtMStM

∣∣∣ Ft]=

1τ>t

(a>r Yt)(a>Yt)

E[(a>r YtM )(a>YtM )

∣∣∣ Ft]=1τ>t

a>ZYt(a>Z 0

)eA(tM−t)

(YtXt

)by applying Lemma A.1. Similarly for contingent cash flows by Lemma A.2 we have

E[e−r(τ−t)1t≤τ≤tM

∣∣∣Gt] =f(τ)1τ>t

StDt

∫ tM

tE[− f(s)DsdSs

∣∣∣ Ft]=

1τ>t

(a>r Yt)(a>Yt)

∫ tM

tf(s)E

[−(a>r Ys)(cYs + γXs)

∣∣∣Ft]ds=1τ>t

a>ZYt

∫ tM

tf(s) a>D eA(s−t)ds

(YtXt

)with f(s) being equal to s or 1, which completes the proof.

Proof of Proposition 5.5

The Levy-Kintchine theorem shows that

Ψ(u) = bZu+

∫ ∞0

(1− e−uξ)νZdξ. (60)

32

We conclude the proof by applying the Sylvester’s formula eUDU−1

= UeDU−1 and by using (60)in (51) as follows

A = bZUDU−1 +

∫ ∞0

(eUDU−1ξ − Id)νZdξ

= bZUDU−1 +

∫ ∞0

(UeDξU−1 − UU−1)νZdξ

= −U(bZ(−D) +

∫ ∞0

(Id−e−(−D)ξ)νZdξ)U−1

= −UΨ(D)U−1.

Proof of Equation (52)

The matrix A in Equation (51) rewrites

A =

∫ ∞0

(eAt − Id)γZt−1e−λZtdt = γZ

∞∑k=1

Ak

k!

∫ ∞0

tk−1e−λZtdt

= γZ

∞∑k=1

Ak

k!

Γ(k)

λkZ= γZ

∞∑k=1

(Aλ−1

Z

)kk

= −γZ log(Id−Aλ−1Z )

where the second equality follows from the definition of the matrix exponential, the third fromthe definition of the Gamma function and its values for integer values, and the last one from thedefinition of the matrix logarithm.

B Market price of risk specifications

We discuss market price of risk (MPR) specifications such that X has a linear drift also under thereal-world measure P ∼ Q. This may further facilitate the empirical estimation of the LHC model.

Let Λ(Yt, Xt) denote the time-t MPR such that the drift of Xt under P becomes

µPt = bYt + βXt + Σ(Yt, Xt)Λ(Yt, Xt).

It is linear in (Yt, Xt) of the formµPt = bPYt + βPXt,

for some vector bP ∈ Rm and matrix βP ∈ Rm×m, if and only if

Λi(y, x) =((bP − b)s+ (βP − β)x)i

σi√xi(y − xi)

, i = 1, . . . ,m. (61)

In order that Λ(Yt, Xt) is well defined and induces an equivalent measure change, that is, thecandidate Radon–Nikodym density process

exp

(∫ t

0Λ(Yu, Xu) dWu −

1

2

∫ t

0‖Λ(Yu, Xu)‖2 du

)(62)

is a uniformly integrable Q-martingale, we need that (Y,X) does not attain all parts of the boundaryof E. This is clarified by the following theorem, which follows from (Cheridito, Filipovic, and Yor2005).

33

Theorem B.1 The MPR Λ(Yt, Xt) in (61) is well defined and induces an equivalent measureP ∼ Q with Radon-Nikodym density process (62) if, for all i = 1, . . . ,m, Xi0 ∈ (0, Y0) and (26)–(27) hold for the Q-drift parameters β, b and for the P-drift parameters βP, bP in lieu of β, b.

If, for some i = 1, . . . ,m, βPij = βij for all j 6= i and

(i) bPi = bi, such that

Λi(y, x) =(βPii − βii)

√xi

σi√y − xi

,

then it is enough if Xi0 ∈ [0, Y0) instead of Xi0 ∈ (0, Y0) and (24) instead of (26) holds forβij , bi, and thus for βPij , b

Pi .

(ii) bPi − bi = βPii − βii, such that

Λi(y, x) =(βPii − βii)

√y − xi

σi√xi

,

then it is enough if Xi0 ∈ (0, Y0] instead of Xi0 ∈ (0, Y0) and (25) instead of (27) holds forβij , bi, and thus for βPij , b

Pi .

The assumption of linear-drift preserving change of measure is often made for parsimony and tofacilitate the empirical estimation procedure. For example, the specification of MPRs that preservethe affine nature of risk-factors has been theoretically and empirically investigated in (Duffee 2002),(Duarte 2004), and (Cheridito, Filipovic, and Kimmel 2007) among others.

C Chebyshev interpolation

This Appendix describes how to perform a Chebyshev interpolation of an arbitrary function on arectangle [a, b]× [c, d] ⊂ R2. The Chebyshev polynomials of the first kind take values in [−1, 1] butcan be shifted and scaled so as to form a basis of [a, b]. In this case they are given by the followingrecursion formula,

ta,b0 (x) = 1

T a,b1 (x) =x− µσ

T a,bn+1(x) =2(x− µ)

σT a,bn (x)− T a,bn−1(x)

with µ = (a+ b)/2 and σ = (b−a)/2. The Chebyshev nodes for the interval [a, b] are then given by

xa,bj = µ+ σ cos (zj) , zj =(1/2 + j)π

N + 1, for j = 0, . . . , N .

The polynomial interpolation of order N is

pN (s, x) =N∑n=0

N∑m=0

cn,m T a,bn (s)T c,dm (x)

where the coefficients are given by

cn,m = 21n 6=0+1m 6=0

N∑i=0

N∑j=0

f(xa,bi , xc,dj

)cos(n zi) cos(mzj)

(N + 1)2.

34

The coefficients can be computed in an effective way by applying Clenshaw’s method, or by ap-plying discrete cosine transform. This straightforward interpolation has the advantage to preventthe Runge’s phenomenon. We refer to (?) for more details on the multidimensional Chebyshevinterpolation, and for an interesting financial application of multivariate function interpolation inthe context of fast model estimation or calibration.

35

all 1 yr 2 yrs 3 yrs 4 yrs 5 yrs 7 yrs 10 yrs

Mean 274.51 144.07 194.80 243.38 279.43 329.40 357.10 373.71Vol 165.23 156.66 158.95 153.31 147.95 141.14 130.46 121.64

Median 244.76 94.79 145.71 189.55 232.44 295.51 353.01 376.58Min 28.02 28.02 39.22 59.50 86.64 109.58 146.32 171.29Max 1288.71 1288.71 1151.92 1092.74 1062.57 1048.33 960.16 887.06

(a) Bombardier Inc.

all 1 yr 2 yrs 3 yrs 4 yrs 5 yrs 7 yrs 10 yrs

Mean 31.01 11.97 17.53 22.74 28.90 34.59 45.00 56.18Vol 21.85 12.93 15.73 17.18 18.18 18.15 16.13 15.66

Median 26.30 7.70 12.42 17.39 24.31 30.45 42.98 55.58Min 1.63 1.63 3.24 4.47 5.81 8.18 12.92 17.51Max 133.02 79.38 102.20 115.19 120.62 126.43 127.22 133.02

(b) Walt Disney Co.

Table 1: CDS spreads summary statistics.The sample contains 552 weekly observations collected between January 1st 2005 and January 1st

2015 summing up to 3620 CDS spreads in basis point for each firm.

36

LHCC(2) LHCC(3) LHCC(3)∗

γ1 0.205 0.201 0.400κ1 0.546 1.263 1.316κ2 0.421 0.668 0.884κ3 0.385 0.668θ1 0.624 0.841 0.696θ2 0.512 0.699 0.548θ3 0.478 0.401

(a) Bombardier Inc.

LHCC(2) LHCC(3) LHCC(3)∗

γ1 0.056 0.064 0.130κ1 0.167 0.258 0.294κ2 0.165 0.229 0.280κ3 0.091 0.212θ1 0.666 0.753 0.558θ2 0.662 0.721 0.536θ3 0.298 0.387

(b) Walt Disney Co.

Table 2: Fitted and fixed (in bold) parameters for the LHC models.

37

all 1 yr 2 yrs 3 yrs 4 yrs 5 yrs 7 yrs 10 yrs

LHCC(2)

RMSE 26.24 23.87 31.79 24.13 12.31 24.36 27.70 33.33Median -0.22 -13.90 -3.16 -1.23 4.63 20.20 -0.17 -18.90

Min -83.96 -64.23 -83.96 -65.09 -22.09 -20.50 -38.64 -79.80Max 123.86 123.86 43.98 32.90 39.31 57.07 75.58 54.45

LHCC(3)

RMSE 16.10 8.90 19.63 19.46 11.01 17.35 15.93 16.94Median -0.25 1.14 -7.69 -5.47 1.06 16.46 2.06 -9.42

Min -56.64 -24.62 -56.64 -52.93 -31.01 -0.66 -12.85 -46.56Max 107.23 107.23 23.86 15.42 20.38 41.61 49.57 31.94

LHCC(3)∗

RMSE 21.87 9.07 23.52 24.01 12.67 16.56 25.15 32.37Median -0.42 0.02 -4.22 -3.94 -3.12 14.22 -0.66 -4.80

Min -82.13 -24.32 -66.96 -68.24 -32.91 -31.95 -54.44 -82.13Max 67.51 24.43 25.10 26.16 22.24 42.51 67.51 59.33

(a) Bombardier Inc.

all 1 yr 2 yrs 3 yrs 4 yrs 5 yrs 7 yrs 10 yrs

LHCC(2)

RMSE 2.88 3.09 1.66 2.73 2.82 2.82 2.00 4.30Median -0.33 -0.13 -0.86 -1.99 -1.40 -0.43 1.40 1.10

Min -12.65 -12.65 -4.15 -5.21 -4.34 -4.32 -5.54 -12.64Max 8.81 3.58 5.11 8.81 8.70 8.22 4.62 6.43

LHCC(3)

RMSE 1.06 0.85 1.09 1.02 0.89 1.31 1.33 0.75Median -0.03 0.35 0.19 -0.55 -0.43 0.14 0.70 -0.26

Min -5.57 -4.87 -5.57 -3.53 -3.55 -4.34 -4.62 -1.97Max 4.94 2.74 4.94 3.58 4.34 3.85 3.53 2.68

LHCC(3)∗

RMSE 1.17 1.02 1.11 0.98 1.15 1.62 1.07 1.12Median 0.01 0.47 0.35 -0.62 -0.60 -0.06 0.48 -0.02

Min -5.48 -5.45 -5.48 -3.49 -3.78 -4.83 -3.92 -4.65Max 4.63 2.68 4.49 3.28 4.63 3.98 2.98 4.15

(b) Walt Disney Co.

Table 3: Comparison of CDS spreads fits for the LHC models.The tables report the minimal, maximal, median, and root mean squared errors in basis point bymaturity over the entire time period for the three different specifications.

38

dS

dX

dX

dX

dS

(1, 0) (1, 1)

(0, 0)

Figure 1: State space of the LHC model with a single factor.Illustrations of the inward pointing drift conditions (24)–(25). The survival process value is givenby the y-axis and the factor value by the x-axis.

39

0 `1 γ

0

3σ

σ

0 γ

0

Figure 2: Comparison of the one-factor LHC and CIR models.Drift and diffusion functions of the default intensity for the one-factor LHC model (black line) andaffine model (grey line). The parameter values are `1 = 0.05, `2 = 1, and γ = 0.25.

40

Jan05 Jan10 Jan15

0

500

1,000

CD

Ssp

read

s

Bombardier Inc.

Jan05 Jan10 Jan15

0

50

100

Walt Disney Co.

−200 0 200 400

−100

0

100

∆1-

year

vs

∆5-

year

−20 0 20

−20

0

20

40

Figure 3: CDS spread data.Panels on the first row display the CDS spreads in basis points for the maturities 1 year (black),5 years (grey), and 10 years (light-grey). Panels on the second row display the weekly changes in1-year versus 5-year CDS spreads.

41

0

500

1,000

Model

spread

s

LHCC(2) LHCC(3) LHCC(3)∗

Jan05 Jan10 Jan150

20

40

60

RMSE

Jan05 Jan10 Jan15 Jan05 Jan10 Jan15

(a) Bombardier Inc.

0

50

100

Model

spread

s

LHCC(2) LHCC(3) LHCC(3)∗

Jan05 Jan10 Jan15

0

2

4

6

RMSE

Jan05 Jan10 Jan15 Jan05 Jan10 Jan15

(b) Walt Disney Co.

Figure 4: CDS spreads fits and errors.Panels on the first row display the fitted CDS spreads in basis points with maturities 1 year(black), 5 years (grey), and 10 years (light-grey) for the three specifications. Panels on the secondrow display the root-mean-square error (in basis points) computed every day and aggregated overall the maturities.

42

1

0.95

0.90St

LHCC(2) LHCC(3) LHCC(3)∗

0

0.10

0.20

λt=γ1X

1t/S

t

Jan05 Jan10 Jan15

0

0.25

0.50

0.75

1

X2t,X

3t

Jan05 Jan10 Jan15Jan05 Jan10 Jan15

(a) Bombardier Inc.

1

0.995

0.990

St

LHCC(2) LHCC(3) LHCC(3)∗

0

0.005

0.010

λt=γ1X

1t/S

t

Jan05 Jan10 Jan15

0

0.25

0.50

0.75

1

X2t,X

3t

Jan05 Jan10 Jan15Jan05 Jan10 Jan15

(b) Walt Disney Co.

Figure 5: Factors fitted from CDS spreads.The filtered factors of the three estimated specifications are displayed over time. Panels on the firstrow display the drift only survival process, panels o the second row the implied default intensity,and panels on the last row the process Xm in black and the process X2 in grey for the three-factormodels.

43

0 500 1000

0

500

1000

1500

Pay

offap

pro

xim

ati

on

k = 250

0 500 1000

k = 300

-500 0 500 1000

k = 350

0 10 20 301

10

100

Err

orb

oun

d

0 10 20 30 0 10 20 30

0 500 1000

0

500

1000

1500P

ayoff

ap

pro

xim

ati

on

k = 250

0 500 1000

k = 300

-500 0 500 1000

k = 350

0 10 20 301

10

100

Err

orb

oun

d

0 10 20 30 0 10 20 30

0 500 1000

0

500

1000

1500

Pay

off

ap

pro

xim

atio

n

k = 250

0 500 1000

k = 300

-500 0 500 1000

k = 350

0 10 20 301

10

100

Err

orb

oun

d

0 10 20 30 0 10 20 30

0 500 1000

0

500

1000

1500

Pay

off

ap

pro

xim

ati

on

k = 250

0 500 1000

k = 300

-500 0 500 1000

k = 350

0 10 20 301

10

100

Err

orb

oun

d

0 10 20 30 0 10 20 30

Figure 6: CDS option payoff approximations.Panels on the first row display the polynomial interpolation of the payoff function approximationwith the Fourier-Legendre approach at the order 1 (light-grey), 5 (grey), and 30 (black). Panels onthe second display the price error bound with the Fourier-Legendre approach (black) and with theChebyshev approach (grey) as functions of the polynomial interpolation order. The first (secondand third) column corresponds to a CDS option with a strike spread of 250 (300 and 350) basispoints. All values are reported in basis points.

44

0 10 20 30200

220

240

k = 250

0 10 20 30

80

100

120

k = 300

0 10 20 30

40

60

k = 350

0 10 20 300.001

0.01

0.1

CPU time

Figure 7: CDS option price approximations and CPU times.The top and bottom-left panels display the price approximation with the Fourier-Legendre approach(black) and with the Chebyshev approach (grey) as functions of the polynomial interpolation order.The top-left (top-right and bottom-left) panels corresponds to a CDS option with a strike spreadof 250 (300 and 350) basis points. All values are reported in basis points. The bottom-right paneldisplays the CPU times in seconds needed to compute the price approximation as functions of thepolynomial interpolation order.

45

0 1 2

0

100

200

σ

CDSO

price

0 0.5 1

0

200

400

600

X0

Figure 8: CDS option price sensitivities.The figure on the left (on the right) displays the CDS option price as a function of the volatilityparameter (the initial risk factor position) for the strike spread 250 (black), 300 (grey), and 350(light-grey). All values are reported in basis points.

46

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